Supergravity

When
supersymmetry is a local symmetry, in addition to chiral and vector
multiplets, there is another multiplet with the graviton and its supersymmetry
partner, the gravitino. Since the graviton has spin 2, the gravitino
Yam, has
spin 3/2, and can be seen in some way as the gauge field of local supersymmetry.
Breaking supersymmetry means giving mass to the gravitino.
As
with a gauge boson, the gravitino can gain mass when the ground state
of the scalar potential breaks the symmetry of the action. In the bosonic
Higgs scenario, the massless Goldstone modes of the scalar field end
up as the extra longitudinal components that make the massless gauge
boson massive. In the supersymmetric case, in addition to Goldstone
bosons, there are massless fermionic
states called Goldstinos, and they provide
the longitudinal modes that give mass to the gravitino and break supersymmetry.
With
supergavity, we have the interesting possibility of breaking supersymmetry
through gravitational couplings. For simple N=1
supergravity with a chiral multiplet, the Kähler potential
looks like

with MP is the Planck mass and W is the superpotential of
the theory. The resulting scalar potential for this theory is

In this model, the gravitino acquires a mass by eating a massless Goldstino,
but because of the minus sign in the scalar potential, the total vacuum
energy can be tuned to be zero. This is important because the total
vacuum energy gives the cosmological constant of the theory, and the
one that has been measured is extremely small.

How to test supersymmetry

One
experimental and theoretical result that is very encouraging evidence
for supersymmetry is the high energy behavior of three Standard Model
coupling constants (two electroweak and one strong). As stated on a
previous page,
the search for a Grand Unified Theory with all Standard Model fields
gathered into representations of one big Lie group was encouraged by
projections that the three Standard Model coupling constants meet at
a single value at some energy scale M = MGUT.
However,
when quantum corrections are included, this agreement does not occur
precisely at a single value. The three coupling constants come much
closer to a single value when the model in which they are being calculated
is the Minimal Supersymmetric Standard Model.
So
supersymmetry suggests unification,
and unification suggests supersymmetry.
None
of this is proof, but it adds a lot of excitement to the search for
proof.
One
thing that a supersymmetric theory should NOT do is violate any of the
observed conservation laws of particle interactions. One important observed
conservation law that is easily violated by unified theories and supersymmetric
theories is the conservation of baryon number.
The
proton is the lightest baryon and hence, if baryon number is conserved,
the proton should be extremely stable. The observed lifetime of the
proton is currently measured to be

Grand
unified theories (GUT for short) have gauge bosons that can mediate
interactions that change quarks into leptons and hence allow the proton
to decay by various interactions, including

where a proton, with baryon number 1, decays into a positron, which
is a lepton and has baryon number 0, and a neutral pion, which is made
of a quark and an antiquark and has baryon number 0. There are three
quarks on the left hand side of the equation and two quarks and a lepton
on the right hand side. If baryon number is not conserved, then the
stable proton becomes unstable. The estimate for the proton lifetime
in a GUT without supersymmetry is

So
this is bad for unification.
In
a GUT with supersymmetry there can also be baryon and lepton number
violation, but for many reasons, the rate ends up being smaller so that

which is still an experimentally viable number, and a region that is
close enough to the observed rate for future measurements, for example,
at at the Super-Kamiokande
experiment in Japan, to be able to tell us something meaningful about
supersymmetry.

Dark matter and SUSY

Because
of the way stars move inside galaxies, astronomers and astrophysicists
have calculated that there is a huge amount of mass in the Universe
that we can't see with telescopes or other instruments because it's
not giving off light the way stars do. That's why they call it dark
matter.
The
presence of this dark matter can be detected by seeing how it interacts
gravitationally, but it's been hard to figure out what it could be made
of. One of the leading candidates for dark matter is a supersymmetry
particle called the LSP, for Lightest Supersymmetric
Particle.
The success of this idea depends on the stability
of the LSP. The LSP is stable in supersymmetric theories with a symmetry
called R-parity, which guarantees that
supersymmetric particles are produced only in pairs. This means that
a supersymmetric particle can only decay into another supersymmetric
particle. Hence the lightest one is stable because it can't decay into
anything.
The LSP that could make up dark matter has to be massive
and electrically neutral, therefore, it could only be the supersymmetry
partner of a neutral particle. The three candidates are: a gravitino
(fermionic superpartner to the graviton), a sneutrino
(scalar superpartner to the neutrino) or a neutralino
(fermionic superpartner to a neutral gauge boson or neutral Higgs scalar).
So far the most promising candidate for dark matter
is the neutralino, because they interact weakly. Therefore they would
decouple from thermal equilibrium at some early age of the universe
and produce a stable residual density that could be large enough to
provide the large amount of dark matter that is believed to be out there.
There
are a lot of hints that supersymmetry could be out there, because it
offers ways to solve many puzzling issues in particle physics and cosmology
at once.
This
is another arrow pointing to string theory, the only theory of elementary
particles that requires both supersymmetry and gravity to exist in Nature.